Theorem lgsdinn0 14116

Description: Variation on lgsdi 14105 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdinn0	|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdinn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5877	. . . . . . . . 9 |- ( x = N -> ( A /L x ) = ( A /L N ) )
2	1	oveq1d 5884	. . . . . . . 8 |- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
3	2	eqeq2d 2189	. . . . . . 7 |- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
4		sq1 10599	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
5	4	eqeq2i 2188	. . . . . . . . . . . . . . . 16 |- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 )
6		nn0re 9174	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> A e. RR )
7		nn0ge0 9190	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> 0 <_ A )
8		1re 7947	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
9		0le1 8428	. . . . . . . . . . . . . . . . . . 19 |- 0 <_ 1
10		sq11 10578	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
11	8, 9, 10	mpanr12 439	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
12	6, 7, 11	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
13	12	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
14	5, 13	bitr3id 194	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) )
15	14	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 )
16	15	oveq1d 5884	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) )
17		1lgs 14111	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> ( 1 /L x ) = 1 )
18	17	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 )
19	16, 18	eqtrd 2210	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 )
20	19	oveq1d 5884	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) )
21		nn0z 9262	. . . . . . . . . . . . . . 15 |- ( A e. NN0 -> A e. ZZ )
22	21	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ )
23		0z 9253	. . . . . . . . . . . . . 14 |- 0 e. ZZ
24		lgscl 14082	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ )
25	22, 23, 24	sylancl 413	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ )
26	25	zcnd 9365	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC )
27	26	mulid2d 7966	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) )
28	20, 27	eqtr2d 2211	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
29		lgscl 14082	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
30	21, 29	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
31	30	zcnd 9365	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC )
32	31	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC )
33	32	mul01d 8340	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 )
34	21	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ )
35		lgs0 14081	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
36	34, 35	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
37		ifnefalse 3545	. . . . . . . . . . . . 13 |- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
38	36, 37	sylan9eq 2230	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 )
39	38	oveq2d 5885	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) )
40	33, 39, 38	3eqtr4rd 2221	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
41		zsqcl 10576	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
42	34, 41	syl 14	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A ^ 2 ) e. ZZ )
43		1z 9268	. . . . . . . . . . . 12 |- 1 e. ZZ
44		zdceq 9317	. . . . . . . . . . . 12 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
45	42, 43, 44	sylancl 413	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ x e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
46		dcne 2358	. . . . . . . . . . 11 |- (DECID ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
47	45, 46	sylib 122	. . . . . . . . . 10 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
48	28, 40, 47	mpjaodan 798	. . . . . . . . 9 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
49	48	ralrimiva 2550	. . . . . . . 8 |- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
50	49	3ad2ant1 1018	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
51		simp3 999	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
52	3, 50, 51	rspcdva 2846	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
53	52	adantr 276	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
54	21	3ad2ant1 1018	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ )
55	54, 23, 24	sylancl 413	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ )
56	55	zcnd 9365	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC )
57	56	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC )
58		lgscl 14082	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
59	54, 51, 58	syl2anc 411	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
60	59	zcnd 9365	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC )
61	60	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC )
62	57, 61	mulcomd 7969	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
63	53, 62	eqtr4d 2213	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) )
64		oveq1 5876	. . . . . 6 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
65	51	zcnd 9365	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
66	65	mul02d 8339	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
67	64, 66	sylan9eqr 2232	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 )
68	67	oveq2d 5885	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
69		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
70	69	oveq2d 5885	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) )
71	70	oveq1d 5884	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) )
72	63, 68, 71	3eqtr4d 2220	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
73		oveq2 5877	. . . . . . . 8 |- ( x = M -> ( A /L x ) = ( A /L M ) )
74	73	oveq1d 5884	. . . . . . 7 |- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
75	74	eqeq2d 2189	. . . . . 6 |- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
76		simp2 998	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
77	75, 50, 76	rspcdva 2846	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
78	77	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
79		oveq2 5877	. . . . . 6 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
80	76	zcnd 9365	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
81	80	mul01d 8340	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
82	79, 81	sylan9eqr 2232	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 )
83	82	oveq2d 5885	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
84		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
85	84	oveq2d 5885	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) )
86	85	oveq2d 5885	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
87	78, 83, 86	3eqtr4d 2220	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
88	72, 87	jaodan 797	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
89		neanior 2434	. . 3 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
90		lgsdi 14105	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
91	21, 90	syl3anl1 1286	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
92	89, 91	sylan2br 288	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
93		zdceq 9317	. . . . 5 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
94	76, 23, 93	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
95		zdceq 9317	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
96	51, 23, 95	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
97		dcor 935	. . . 4 |- (DECID M = 0 -> (DECID N = 0 -> DECID ( M = 0 \/ N = 0 ) ) )
98	94, 96, 97	sylc 62	. . 3 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
99		exmiddc 836	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
100	98, 99	syl 14	. 2 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
101	88, 92, 100	mpjaodan 798	1 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   ifcif 3534   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   x. cmul 7807   <_ cle 7983   2c2 8959   NN0cn0 9165   ZZcz 9242   ^cexp 10505   /Lclgs 14065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066

This theorem is referenced by: (None)